Analytical results for the unusual Gr\"uneisen ratio in the quantum Ising model with Dzyaloshinskii-Moriya interaction

TL;DR

This paper investigates the behavior of the Grüneisen ratio in a quantum Ising model with Dzyaloshinskii-Moriya interaction, revealing distinct divergence patterns at different phase transitions and challenging existing paradigms.

Contribution

It provides an exact analysis of the Grüneisen ratio in a solvable quantum Ising model with Dzyaloshinskii-Moriya interaction, highlighting its behavior at various quantum phase transitions.

Findings

GR remains finite at the ferromagnetic--paramagnetic transition despite broken self-duality.

GR diverges and changes sign at the transition between gapped ferromagnetic and gapless Luttinger liquid phases.

GR can probe the nature of first-order versus continuous quantum phase transitions.

Abstract

The Gr\"uneisen ratio (GR) has emerged as a superb tool for the diagnosis of quantum phase transitions, which diverges algebraically upon approaching critical points of continuous phase transitions. However, this paradigm has been challenged recently by observations of a finite GR for self-dual criticality and divergent GR at symmetry-enhanced first-order transitions. To unveil the fascinating GR further, we exemplify the idea by studying an exactly solvable quantum Ising model with Dzyaloshinskii-Moriya interaction, which harbors a ferromagnetic phase, a paramagnetic phase, and a chiral Luttinger liquid. Although the self-dual criticality of the ferromagnetic--paramagnetic transition is undermined by the Dzyaloshinskii-Moriya interaction, we find that the GR at the transition is still finite albeit with an increasing value, signifying a proximate self-dual relation. By contrast, the GR at the transition between the gapped ferromagnetic phase and the gapless Luttinger liquid diverges and changes its sign when crossing the first-order transition. This implies that the GR could also probe the first-order transition between the gapped and gapless phases.